Fibrations on the 6-sphere

Speaker: 

Jeff Viaclovsky

Institution: 

UC Irvine

Time: 

Tuesday, May 21, 2024 - 4:00pm to 5:00pm

Host: 

Location: 

ISEB 1200

Let be a compact, connected 3-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from Z onto any 2-dimensional complex space. Combining this with a result of Campana-Demailly-Peternell, a corollary is that any holomorphic mapping from the 6-dimensional sphere S^6, endowed with any hypothetical complex structure, to a strictly lower-dimensional complex space must be constant. In other words, there does not exist any holomorphic fibration on S^6. This is joint work with Nobuhiro Honda.

Free boundary minimal surfaces via Allen-Cahn equation

Speaker: 

Martin Li

Institution: 

Chinese University of Hong Kong

Time: 

Tuesday, April 30, 2024 - 4:00pm

Host: 

Location: 

ISEB 1200

It is well known that the semi-linear elliptic Allen-Cahn equation arising in phase transition theory is closely related to the theory of minimal surfaces. Earlier works of Modica and Sternberg et. al in the 1970’s studied minimizing solutions in the framework of De Giorgi’s Gamma-convergence theory. The more profound regularity theory for stationary and stable solutions were obtained by the deep work of Tonegawa and Wickramasekera, building upon the celebrated Schoen-Simon regularity theory for stable minimal hypersurfaces. This is recently used by Guaraco to develop a new approach to min-max constructions of minimal hypersurfaces via the Allen-Cahn equation. In this talk, we will discuss about the boundary behaviour for limit interfaces arising in the Allen-Cahn equation on bounded domains (or, more generally, on compact manifolds with boundary). In particular, we show that, under uniform energy bounds, any such limit interface is a free boundary minimal hypersurface in the generalised sense of varifolds. Moreover, we establish the up-to-the-boundary integer rectifiability of the limit varifold. If time permits, we will also discuss what we expect in the case of stable solutions. This is on-going joint work with Davide Parise (UCSD) and Lorenzo Sarnataro (Princeton). This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China. 

Special Seminar in Geometric Analysis

Institution: 

Special Seminar

Time: 

Tuesday, May 28, 2024 - 2:00pm to 5:45pm

Location: 

ISEB 1010

Talk Schedule (each talk 30-35 minutes)

2:00 PM   Alex Mramor (University of Copenhagen)

2:45 PM   Kai-Wei Zhao (Notre Dame)

3:30 PM   Hongyi Sheng (UC San Diego)

4:15 PM   Tin Yau Tsang (New York University)

5:00 PM   Xiaolong Li (Wichita State University)

 

Titles/Abstracts

Speaker: Alex Mramor (University of Copenhagen)
Title: On the Unknottedness of Self Shrinkers
Abstract: The mean curvature flow, the natural analogue of the heat equation in submanifold geometry, often develops singularities and roughly speaking these singularities are modeled on self shrinkers, which are surfaces that give rise to mean curvature flows that move by dilations. it happens that self shrinkers are minimal surfaces in a metric which, while poorly behaved, is Ricci positive in a certain sense so it is natural, for instance, to ask what type of qualities shrinkers have in common with minimal surfaces in the round 3-sphere. Inspired by an old work of Lawson on such surfaces in this talk we discuss some unknottedness results for self shrinkers in R3, some of which are joint work with S. Wang.

 

Speaker: Kai-Wei Zhao (Notre Dame)
Title: Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves
Abstract: Curve shortening flow is, in compact case, the gradient flow of arc-length functional. It is the simplest geometric flow and is a special case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions can be view as a parabolic analogue of geometric Liouville theorem. The previous results technically reply on the assumption of convexity of the curves. In the ongoing project joint with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace it by the boundedness of entropy, which is a measure of geometric complexity defined by Colding and Minicozzi. In this talk, we will prove that an ancient smooth curve shortening flow with finite-entropy embedded in R2 has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity m≥3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.

 

Speaker: Hongyi Sheng (UC San Diego)
Title: Localized Deformations and Gluing Constructions in General Relativity
Abstract: Localized deformations play an important role in gluing constructions in general relativity. In this talk, we will review some recent localized deformation theorems and their applications regarding rigidity and non-rigidity type results.

 

Speaker: Tin Yau Tsang (New York University)
Title: Mass for the Large and the Small
Abstract: The positive mass theorem concerns the mass of large manifolds. In this talk, we will first review the proofs by Schoen and Yau, then the proof by Witten. Combining these with their recent generalisations turns out to help us understand the mass of small manifolds.

 

Speaker: Xiaolong Li (Wichita State University)
Title: Recent Developments on the Curvature Operator of the Second Kind
Abstract: In this talk, I will first introduce the curvature operator of the second kind and talk about the resolution of Nishikawa's conjecture by Cao-Gursky-Tran, myself, and Nienhaus-Petersen-Wink. Then I will talk about some ongoing research with Gursky concerning negative lower bounds of the curvature operator of the second kind. Along the way, I will mention some interesting problems.

How rare are simple Steklov eigenvalues?

Speaker: 

Lihan Wang

Institution: 

CSU Long Beach

Time: 

Tuesday, February 27, 2024 - 4:00pm

Location: 

ISEB 1200

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

The roles of concavity, symmetry and sub-solutions in geometric PDEs

Speaker: 

Bo Guan

Institution: 

Ohio State University

Time: 

Tuesday, January 30, 2024 - 4:00pm to 5:00pm

Location: 

ISEB 1200

In this talk we discuss the roles of concavity, symmetry and subsolutions in the study of fully nonlinear PDEs, especially those on real or complex manifolds with connection to geometric problems. We shall report some of our results along the line, which give the optimal conditions for the existence of classical solutions, either of the Dirichlet problem, or of equations on closed manifolds. If time permits, we shall also discuss the possibility to weaken or extend these conditions, and a class of equations involving differential forms of higher rank, more specifically real (p, p) forms for p > 1 on complex manifolds. Part of the talk is based on joint work with my student Mathew George.

A free boundary problem in pseudoconvex domains

Speaker: 

Chi Fai Chau

Institution: 

UC, Irvine

Time: 

Tuesday, November 28, 2023 - 4:00pm

Host: 

Location: 

ISEB 1200

A domain with C^2 boundary in complex space is called pseudoconvex if it has a C^2 defining function with positive complex hessian on its boundary. Pseudoconvexity is a generalization of convexity. It can be realised as a domain with geometric condition on the boundary and its topology can be studied by Morse theory. In this talk, we will discuss the Morse index theorem for free boundary minimal disks for partial energy in strictly pseudoconvex domain and the relation between holomorphicity and stability of the free boundary minimal disk. We will also give an example to illustrate the necessity of strict pseudoconvexity in our index estimate.

On complete Calabi-Yau manifolds asymptotic to cones

Speaker: 

Junsheng Zhang

Institution: 

UC Berkeley

Time: 

Tuesday, November 14, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

ISEB 1200

We proved a ``no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate  to the asymptotic cone for such manifolds. Joint work with Song Sun.

 

The parabolic U(1)-Higgs equations and codimension-two mean curvature flows

Speaker: 

Davide Parise

Institution: 

UC San Diego

Time: 

Tuesday, October 3, 2023 - 4:00pm

Location: 

ISEB 1200

Mean curvature flow is the negative gradient flow of the area
functional, and it has attracted a lot of interest in the past few years. In
this talk, we will discuss a PDE-based, gauge theoretic, construction of
codimension-two mean curvature flows based on the Yang-Mills-Higgs
functionals, a natural family of energies associated to sections and metric
connections of Hermitian line bundles. The underlying idea is to approximate
the flow by the solution of a parabolic system of equations and study the
corresponding singular limit of these solutions as the scaling parameter
goes to zero. This is based on joint work with A. Pigati and D. Stern.

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